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gap> nilp := SchunckClass(rec(bound := G -> not IsCyclic(G),
> name := "class of all nilpotent groups"));
class of all nilpotent groups
gap> DihedralGroup(8) in nilp;
true
gap> SymmetricGroup(3) in nilp;
false
gap> H := SchunckClass(rec(bound := G -> Size(G) = 6));
SchunckClass(bound:=function( G ) ... end)
gap> Size(Projector(GL(2,3), H));
16
gap> # H-projectors coincide with Sylow subgroups
gap> U := SchunckClass(rec( # class of all supersoluble groups
> bound := G -> not IsPrimeInt( Size(Socle(G)))
> ));
SchunckClass(bound:=function( G ) ... end)
gap> Size(Projector(SymmetricGroup(4), U));
6
gap> # the projectors are the point stabilizers
gap> der3 := OrdinaryFormation(rec(
> res := G -> DerivedSubgroup(DerivedSubgroup(DerivedSubgroup(G)))
> ));
OrdinaryFormation(res:=function( G ) ... end)
gap> SymmetricGroup(4) in der3;
true
gap> GL(2,3) in der3;
false
gap> exp6 := OrdinaryFormation(rec(
> \in := G -> 6 mod Exponent(G) = 0,
> char := [2,3]));
OrdinaryFormation(in:=function( G ) ... end)
gap> nilp := SaturatedFormation(rec(
> locdef := function(G, p)
> return GeneratorsOfGroup(G);
> end));
SaturatedFormation(locdef:=function( G, p ) ... end)
gap> form := SaturatedFormation(rec(
> locdef := function(G, p)
> if p = 2 then
> return GeneratorsOfGroup(G);
> elif p mod 4 = 3 then
> return GeneratorsOfGroup(DerivedSubgroup(G));
> else
> return fail;
> fi;
> end));
SaturatedFormation(locdef:=function( G, p ) ... end)
gap> Projector(GL(2,3), form);
Group([ [ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3)^0, Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ])
gap> nilp := SaturatedFormation(rec(\in := IsNilpotent, name := "nilp"));
nilp
gap> FormationProduct(nilp, der3); # no characteristic known
FormationProduct(nilp, OrdinaryFormation(res:=function( G ) ... end))
gap> HasIsSaturated(last);HasCharacteristic(nilp);
false
false
gap> SetCharacteristic(nilp, AllPrimes);
gap> FormationProduct(nilp, der3); # try with characteristic
FormationProduct(nilp, OrdinaryFormation(res:=function( G ) ... end))
gap> IsSaturated(last);
true
gap> nilp := FittingFormation(rec(\in := IsNilpotent, name := "nilp"));;
gap> FormationProduct(nilp, nilp);
FittingFormationProduct(nilp, nilp)
gap> FittingProduct(nilp, nilp);
FittingFormationProduct(nilp, nilp)
gap> FittingFormationProduct(nilp, nilp);
FittingFormationProduct(nilp, nilp)
gap> G := DirectProduct(SL(2,3), CyclicGroup(2));;
gap> data := rec(gens := GeneratorsOfGroup(G),
> comms := List(Combinations(GeneratorsOfGroup(G), 2),
> x -> Comm(x[1],x[2])));;
gap> OneInvariantSubgroupMinWrtQProperty(
> G, G,
> function(U, V, R, data) # test if U/V is central in G
> if ForAny(ModuloPcgs(U, V), y ->
> ForAny(data.gens, x -> not Comm(x, y) in V)) then
> return false;
> else
> return fail;
> fi;
> end,
> function(S, R, data)
> return ForAll(data.comms, x -> x in S);
> end,
> data) = DerivedSubgroup(G); # compare results
true
gap> G := GL(2,3);
GL(2,3)
gap> normsWithSupersolubleFactorGroups :=
> AllInvariantSubgroupsWithQProperty(G, G,
> function(U, V, R, data)
> return IsPrimeInt(Index(U, V));
> end,
> ReturnFail, # pretest is sufficient
> fail); # no data required
[ GL(2,3),
Group([ [ [ Z(3)^0, Z(3) ], [ 0*Z(3), Z(3)^0 ] ],
[ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ],
[ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]),
Group([ [ [ Z(3), Z(3)^0 ], [ Z(3)^0, Z(3)^0 ] ],
[ [ 0*Z(3), Z(3)^0 ], [ Z(3), 0*Z(3) ] ],
[ [ Z(3), 0*Z(3) ], [ 0*Z(3), Z(3) ] ] ]) ]
[ Dauer der Verarbeitung: 0.25 Sekunden
(vorverarbeitet)
]
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